Game-theoretic probability

Abstract

The standard theory of probability is based on Kolmogorov's measure-theoretic axioms. A less known alternative is the game-theoretic approach to probability. The purpose of this chapter is to give an introduction to the current state of game-theoretic probability. The chapter begins by stating a simple game-theoretic strong law of large numbers. This motivates the introduction of a general discrete-time forecasting protocol and the definition of game-theoretic expectation and probability. The chapter discusses the axiom of continuity for sets of available gambles, and the Doob's argument, which is useful in measure-theoretic and game-theoretic probability. Some limit theorems of game-theoretic probability are also outlined in the chapter. The chapter discusses a different kind of classical results of probability, the zero-one laws, in particular, the Lévy's zero-one law. The last section gives references for further reading.